Chapter 6: Beyond Upper Hessenberg Form

Jared L. Aurentz, Thomas Mach, Leonardo Robol, Raf Vandebril, David S. Watkins

2018

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DOI 10.1137/1.9781611975345.ch6
Data confidence 70 medium
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Abstract

Up to now we have restricted our attention to upper Hessenberg matrices. In this chapter we will introduce a large family of condensed forms that are generalizations of upper Hessenberg form, and we will develop generalizations of Francis's algorithm that operate on them. In [64] we have shown that the performance of the algorithm can be affected by the choice of condensed form, and since the form can evolve over the course of iterations, there is the prospect of building a faster algorithm that chooses the condensed form adaptively. Although the realization of this prospect lies in the future, we are pleased to include this new theory here because we find it really interesting, and we hope the reader will too. The material in this chapter is drawn mainly from [62] and [64], but see also [63].